Chapter1Remainder and FactorThm(S)
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Transcript of Chapter1Remainder and FactorThm(S)
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At the end of the lesson, students should
be able to:
use the remainder and factor theorems.
LEARNING OUTCOMES
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Consider
The remainder may be
obtained by long division
as follows
2267 23 xxxx45
10
84
24
105
65
2
2672
2
2
2
23
23
xx
x
x
xx
xx
xx
xxxx
10245267 223 xxxxxx
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Remainder
Alternatively,
2P
10
02x 2xLet
2267 23 xxxx
22627223
267 23 xxxxPLet
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DEFINITIONWhen a polynomial P(x)
is divided by a linear
factor (x-a), then the
remainder is P(a)
Remainder
Theorem
Proof:
P(x) = (x-a) Q(x) + R(x)
When x= a;
P(a) = 0 + R(x)
= R(x)
P(x) = D(x) Q(x) + R(x)
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EXAMPLE
By using remainder theorem, find the
remainder when is
divided by
62 23 xxxP
3x
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By remainder theorem
Remainder
SOLUTION
3P
632323
39
03x 3xLet
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YOUR TURN!!
Using remainder theorem, find the remainder for
the following
i.
ii.
iii.
iv.
2 ; 137 23 xxxxxP
xxxxP ; 122
2 ; 232 2 xxxxP
1 ; 534
xxxP
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When a polynomial P(x) = x3 + px2 + qx -9
is divided by x+1, the remainder is 3 and
when P(x) is divided by x-2, the remainder is
9. Find the value of p and q.
EXAMPLE
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When is divided by
(x + 2), the remainder is 16. Determine k.
865 234 xxkxx
EXAMPLE
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Let
Given that
865)( 234 xxkxxxP
16)2(P
168)2(6)2(5)2()2( 234 k
248k
3k
SOLUTION
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a) Given that .
When P(x) is divided by , the
remainder is twice of the remainder
when P(x) is divided by . Find a .
EXERCISE
162)( 23 xaxxxP
2x
1x
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EXAMPLE
By using remainder theorem, find the
remainder when is
divided by
276 2 xxxP
12x
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If the remainder obtained from dividing the
polynomial P(x) by (x-a) is zero, then the linear term
(x-a) is called a factor of the polynomial P(x).
If P(a) = 0 then (x a) is a factor of P(x)